As multi-dimensional beings, humans have long felt a need to map out and measure their surroundings. Because of this, various coordinate systems, including Cartesian, polar, cylindrical, and spherical coordinate systems, have been developed, and are widely used in everyday life to help designate where in space an object is in relation to a reference point, or in relation to another object.
The polar coordinate system, for instance, is a two-dimensional coordinate system wherein each point on a plane is determined by a distance from a reference point and an angle from a reference line having a fixed direction, the reference point being the vertex of the angle formed by the reference line and the angle. The reference point (analogous to the origin of a Cartesian system) is defined as the pole, and the line having the fixed direction is defined as the polar axis. The distance from the pole to the point in space may be referred to as the radial coordinate or radius, and the angle may be referred to as the angular coordinate, polar angle, or azimuth.
A cylindrical coordinate system is a three-dimensional coordinate system that provides for a determination of a position of a point using the distance from a chosen reference axis, the direction from the axis relative to a chosen direction, and the distance from a reference plane perpendicular to the axis. The distance from the chosen reference axis may be provided as a positive or negative number depending on which side of the reference plane faces the point for which the position is to be determined. The intersection between the reference plane and the axis is defined as the origin of the system. More specifically, the origin of the system may be defined as the point where all three coordinates may be given a zero value. The axis may be referred to as the cylindrical or longitudinal axis, to differentiate it from the polar axis. The distance from the axis to the point in space may be referred to as the radial distance or radius, while the angular coordinate may sometimes be referred to as the angular position or as the azimuth. The radius and the azimuth are therefore defined as the polar coordinates, as they correspond to a two-dimensional polar coordinate system in the plane through the point, parallel to the reference plane. The third coordinate may be defined as the height or altitude (if the reference plane is considered horizontal), longitudinal position, or axial position. Cylindrical coordinates are useful in connection with measurements relating to objects and phenomena that have some rotational symmetry about the longitudinal axis, such as water flow in a straight pipe with a round cross-section, heat distribution in a metal cylinder, and so on.
A spherical coordinate system may be described as a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its inclination angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. The inclination angle is often replaced by the elevation angle measured from the reference plane. The radial distance may also be referred to as the radius or radial coordinate, and the inclination angle may also be referred to as co-latitude, zenith angle, normal angle, or polar angle.
In geography and astronomy, the elevation and azimuth (or quantities very close to them) may be referred to as the latitude and longitude, respectively. Radial distance may be replaced by an altitude (measured from a central point or from a sea level surface). Spherical coordinates can also be used in relation to measurements extending to higher dimensional spaces and, in such a case, may be referred to as hyper-spherical coordinates. One of the disadvantages of the above referenced coordinate systems, however, is the inability to measure both of the angles that define a tridimensional position, i.e., two angles orthogonal to each other within the same plane. Such a situation proves difficult for a user to measure both angles in a unique observer's position.
Accordingly, a need exists for an angular measurement system and method for using an angular measurement device that may advantageously be used to read angles within orthogonal vertical planes that can be used to infer three-dimensional angles.